In-plane vibrations of circular rings on a tensionless foundation

Journal ofSound

and Vibration (1990) 143(3), 461-471

IN-PLANE VIBRATIONS OF CIRCULAR RINGS ON A TENSIONLESS FOUNDATION Z.

Faculty of Civil Engineering,

(Received

4 September

CELEP

Technical University, 80626 Maslak, Istanbul, Turkey 1989, and in revised form 21 February 1990)

The problem of a thin elastic circular ring on a tensionless Winkler foundation is considered. It is assumed that the ring is subjected to time dependent in-plane loads. The analysis is based on the harmonic approximation for the deflection function of the ring. The governing equations of the problem are derived by employing Lagrange’s equations. The static and dynamic responses of the ring are obtained numerically and presented in figures comparatively for conventional and for tensionless Winkler foundations.

1. INTRODUCTION

Rings are important elements in structures such as gears, electrical machines and stiffened shells. The vibrations of complete rings have therefore been the subject of many investiga-

tions. The classical solution for the free in-plane vibrations of a thin ring, with neglect of the effects of extensibility of the middle circle, and shear and rotatory inertia contributions was given by Hoppe [l]. The classical solution has been improved by including the shear and rotatory inertia effects and it has been shown by several authors that the effect of extension is quite negligible. Of the numerous studies, the papers of Nelson [2], Seidel and Erdelyi [3], and Rao and Sundararajan [4] are of interest, in which approximate solutions techniques were employed. There are a number of other parameters which complicate the problem, such as non-uniformity of the cross-section, which has been considered by Filipich et al. [5], and the presence of an elastic foundation, which was studied by Rao [6]. Further, Mittal [7] investigated the problem of a complete ring subjected to a time dependent concentrated load in its plane. When rings are used as connection elements between two structures to transfer forces, they may be modelled as rings supported elastically on structural parts of cylindrical shape. The subject of the present paper is the response of an elastic thin ring supported by a Winkler foundation. In the conventional foundation model it is assumed that the foundation reacts in compression as well as in tension. However, there are many cases in which the admission of tensile stresses across the contact region which separates the structure from the foundation is questionable. From this point of view, the circular ring is assumed to be supported on a foundation that reacts in compression only. In this foundation model, instead of the tensile stresses, a lift-off comes into being and a gap appears between the ring and the foundation. However, the response of the ring is complicated by the need to determine the contact region as a function of time for the dynamic loading cases. Probably due to this mathematical difficulty, problems dealing with a tensionless foundation have received only limited attention. Recently, a number of such problems were studied by considering a beam, and circular and rectangular plates on tensionless foundations [g-14]. In the present paper the solution technique is extended to a ring which undergoes in-plane vibrations. 461 0022-460X/90/240461 + 11 $03.00/O

@ 1990 Academic hess Linited

462

2. 2. BASIC

CELEP

EQUATIONS

Consider a thin elastic ring of radius R resting on a tensionless radial and tangential Winkler foundation of modulus k, and k,, respectively, as shown in Figure 1. The ring is assumed to be subjected to a distributed radial load pI( 0, t) and a distributed tangential load p,( 0, t), which are functions of position and time. In view of the difficulties in finding an exact solution for the radial and tangential displacement functions, u( 0, t) and u( 8, t),

qy!$&J @.JtJ p$iJ (d)

(e)

(f)

Figure 1. Circular ring on a tensionless Winkler foundation.

which satisfy the governing equations and the boundary conditions of the problem, the tangential displacement is assumed to be of the form ~(~,f)=Ru~(t)+R

f

n=,

u,,(t)cosnt3+R

: ~,,(t)sinn@, I?=1

(1)

and the radial displacement is obtained, by using the condition of inextensibility of middle circle of the ring, as u( e, t) = u’( 0, t) = R z m,,,(t)

n=,

cos no -

R z

fl=l

TV,,

sin ne.

(2)

The solution is sought by employing Lagrange’s equation (d/dt) aT/a&

+aU/aq,

=O,

(3)

with qn = uo, v,, and v,,. The kinetic energy of the ring including both the radial and the tangential displacement contributions is expressed as 277

m(ti2+ti2)R de, I0 and the total potential energy of the ring including the potentials tangential external loads, pr and p,, is given as 7-z;

U=

I

02w$$

(u”+ v’)~R de+;

H(0)[k,.u2+

(4) of the radial and

k,u’]R de

27r (P,~+PP)R -I

0

d’$

(5)

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where EZ is the bending stifhress of the cross-section and m is the mass per unit arc length of the ring. The prime and dot denote derivatives with respect to 0 and t, respectively. The bending moment acting on the ring is given as M(f3, t) = (Ez/RZ)(u”+v’). The tensionless character of the foundation function H(B, r) which is defined as H(6, t)=

(6)

is taken into account by using an auxiliary for ~(0, t)>OJ for ~(0, t)SOJ

:,

(7)

This function keeps track of the contact status of the ring. For the conventional Winkler foundation, one has to assume H( 8, t) = 1 irrespective of the sign of u( 0, t). By substituting the displacement functions into equations (4), (5) and (6), the following system of differential equations is obtained:

uoao+; (a,u,,+b,u,,) ItI=

i;,, +

uo+

PI de,

0

1)2

n'(n2-

k;a,

7r(n2+ 1)

1 I

27r

=&

U”C

n2+1

+nmk;d,,)u,,+(~,c,,-nm~~c,,)o,,l =

i& +

(~,COS

k;bn x(n2+ 1)

uo+

ne-q,

n2(n2-1)2 n2+1

sin

ne)

de,

u?l,

2t.r

1 = 7r(n2+1)

I0

(p,

sin

ne+q,

(8)

cos ne)de.

Here 2T a,

=

2n

H(~,T)cos

ne de,

I0

b, =

H( 8, T) sin no de, I

277

2n

H( 8, 7) cos me sin ne de,

CIn"=

0

I0

H( 8,~) sin me sin ne de,

d,, = I0

2m

H( 8,~) cos me cos n0 de,

emn =

(9)

I0

and the non-dimensional k; = k,R4/ EZ, The dots in equations

foundation

constants and loads are

6, = k,R4/ EZ,

j& = p,R’/ El,

j& = p,R3/ EI.

(8) denote derivatives with respect to the non-dimensional

(10)

time r

defined as r=tm/R’.

(11)

464

Z. CELEP

Equations (8) can be written in a matrix form as i;+Kv=f,

(12)

where vT= (DO,UlC,uzc, ’. . , Uls, vzs, . . *). The matrix K is the stiffness matrix and f is the loading vector of the ring. Although the present formulation does not have any restriction concerning the external loading, the right sides of equations (8), in dimensional form, cos nf3 - np, sin no) R de,

p,R de,

( pr sin no + .np, cos f9)R de, (13)

can be evaluated as follows for the loading cases shown in Figure 1: for the singular radial load Q. (Figure l(a)), -nQo sin ne,,

0,

nQo cos ne,;

(14)

for the singular vertical load W. (Figure l(b)), W. sin Oo,

W, sin e. cos neo - n W, cos e. sin ne,,

W, sin f30sin no,+ n W, cos e. cos ne,;

(15)

for the singular tangential load PO (Figure l(c)), P 0,

for the uniformly distributed

for the uniformly distributed e,-c0s

ne,,

POsin ne,;

(16)

radial load q. (Figure 1(d)),

qoR(cos ne2-cos

0,

-W,R(COS

POcos

no,),

q,R(sin no,-sin

no,);

(17)

vertical load w. (Figure l(e)),

e,),

i)e,-c0s(n+l)e,] n+l

0*5woR n-_1[sin(n-1)82-sin(n-1)8,]+~[sin(n+1)8,-sin(n-1)8,]

for the uniformly distributed poR(e2-

e,),

I ,

(18)

tangential load p. (Figure l(f )),

p,R(sin no,-sin

-p,R(cos

n&)/n,

ne,-cos

n&)/n.

(19)

Due to the tensionless character of the foundation, the stiffness matrix K depends on the loading configuration and the contact region; thus the problem involves the solution of a non-linear differential equation system. The static behaviour of the ring can be obtained by solving v = K-‘f. The diffi cu Ity in solving the static as well as the dynamic problem is due to the fact that the contact region, and consequently, the contact function H( 0, r) are not known in advance and depend on the loading.

3. NUMERICAL

RESULTS

AND

DISCUSSION

Numerical calculations were carried out at the Computer Center of the Technical University, Istanbul. The non-linear static problem was solved numerically by using an

RING

ON TENSIONLESS

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FOUNDATION

iterative scheme. The integrals (9) were evaluated by using the Gauss-Legendre quadrature [ 151 and the solution of the dynamic problem was carried out by employing the RungeKutta method [16]. Since the differential equation is solved step by step, the contact region of the ring and consequently the function H(B, t) is updated at each time step in the course of the numerical solution. For the number of the integration points in the integrals (9), the number of the terms considered in the series (1) and (2) and for the length of the time step, several test cases were calculated and compared. As a result of these comparisons, 48 integration points and 10 terms in the series were taken into account to obtain accurate graphical representation of the numerical results. In all numerical calculations k, = 0 is assumed. The top and the bottom displacements of the ring subjected to a uniformly distributed vertical load are shown in Figure 2. The results are illustrated for the tensionless and the conventional foundation models as functions of the foundation stiffness. The figure also displays the contact angle which separates the contact and the lift-off regions for the tensionless foundation and the compression and the tension regions for the conventional foundation. When the foundation is very soft, the rigid displacement of the ring becomes dominant. Consequently, the top and the bottom displacements are almost equal and the contact angle &, is 7r/2. Since half of the ring lifts off the foundation, a marked difference in the displacements comes into being for the tensionless and the conventional foundations. For a very low foundation stiffness where the rigid displacement is pronounced, the value of k,ul w. approaches 2 and 4 for the conventional and the tensionless cases, respectively. As the foundation becomes stiffer, the non-dimensional displacement and the contact angle do not show any marked variation for the conventional case. However, when the foundation is assumed to be tensionless the ring sags significantly, and the contact angle increases as the foundation gets stiffer whereas the displacement of the top point of the ring varies slowly. This figure represents clearly how the behaviour of the ring changes for the tensionless and conventional foundation models.

Figure 2. Top and bottom displacements and contact angle of the ring subjected vertical load; tensionless and - - - conventional Winkler foundation.

to a uniformly

distributed

466

Z. CELEP

The displacements and the contact angle for the uniformly distributed vertical and radial loads are shown in Figures 3 and 4, respectively. The curves which correspond to the conventional case are also drawn for the singular loading, but only for the purpose of comparison. For the tensionless case various distribution angles are considered by assuming the total load to be constant. As can be seen, the effect of the loading angle on the response of the ring becomes greater as the foundation becomes stiffer. For a very soft foundation contact is established on the upper half of the ring, whereas the lower half lifts off the foundation. In this case the ring experiences almost a rigid displacement.

90

4

??

3

2 ‘;o a7 N r i 1

0

0

2

4

0

2

4

log L Figure 3. Top and bottom displacements radial loads. Key as Figure 2.

of the ring subjected to uniformly distributed (a) vertical and (b)

Consequently, good accuracy can be obtained by considering very few terms in the series expansions of the displacements. The limiting values of the displacements can be obtained for k, = 0 and kr + 0 by considering a rigid displacement u( 6) = u. cos 8 and by writing the global force equilibrium of the ring for the tensionless foundation model as 2uok

cos28 de = 24wo

2q, jo”lcos 8 d0

(20)

for the vertical and radial loading cases, respectively. This leads to Iim [k,u,/(28,wo)]=2/r,

lim [k,u,/(28,qJ]

k,+O

k,+O

= 2 sin e,/(&,).

(21)

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ON TENSIONLESS

467

FOUNDATION

160

135

c : B 0 90 D 0”

45

Figure 4. Contact as Figure 2.

angle of the ring subjected

to uniformly

distributed

(a) vertical

and (b) radial

loads.

Key

Since total contact is assumed for the conventional case, the corresponding limiting values are half of those for the tensionless case. Comparison of the results in the figures with these limiting values reveals a very good agreement. As is seen in the figures, the ring experiences two contact regions for the tensionless case at its top and at its bottom parts and these two regions approach to each other as the foundation becomes stiffer. Although the formulation allows one to consider any arbitrary time variation of the external load, for the sake of brevity, it is assumed that vibrations of the ring are caused by a sudden change in the static load factor p, corresponding to unloading or loading when it is smaller or larger than one, respectively. As is seen in Figure 5 an adequate

0

4 0.0 Figure

5. Time variations

0.1

0.2

of the top displacement

0.3

0.4

of the ring for various

0.5 time-step

0.6 lengths;

L, = 1000, p = 0.5.

Z. C‘ELEP

468

accuracy can be achieved by using a time step AT SO.025.The time variations of the displacements of the ring for various values of the loading factor p for the conventional and the tensionless models are shown in Figures 6(a) and (b), respectively. Since the conventional foundation is relatively more constrained, the displacements are small in -2 r

-1

in

3 0.0

90

((1)

‘v' 600

0.5

I.0

T

1.5

2.0

Figure 6. Time variations of the displacements of the ring subjected to a radial and (b) the tensionless foundation; k; = 1000. 0, /3 = 0; 0, p = 0.5; A., p = 2.

2.5

load on (a) the conventional

comparison with those for the tensionless case. In the conventional case the free vibration modes can be defined as usual and the forced vibration modes can be expressed as combinations of these modes. A certain type of combination appears in the response of the ring given in Figure 6(a). Here the dominant vibration period also can be easily seen. On the other hand, the variations for the tensionless case are shown in Figure 6(b). As is seen, the response differs from that of the conventional case markedly and the oscillations having large amplitudes appear to be much more complex. It is difficult to recognize a dominant period in the oscillations due to the non-linearity of the problem. The time variations of the displacements for the unloading and loading cases, respectively, when the ring is subjected to a singular load, are shown in Figures 7(a) and 7(b). The oscillations appear to be complex for both cases. This is due to the fact that the singular load excites vibration modes of higher order than does the distributed load. However, large displacements appear in a more marked manner in the tensionless case.

RING

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1

0.0

I

0.5

469

FOUNDATION

1.0

I

,

1.5

2.0

2.5

I-

Figure 7. Time variations of the top displacement of the ring subjected to a singular load for (a) unloading and (b) loading cases; 5 = 1000. For (a): 0, p = 0; 0, p = 0.5; for (b): 0, 0 = 1.5; 0, /I = 2; tensionless and - - - conventional Winkler foundation.

I

00

0.5

Figure 8. Time variations of the displacements k; = 1000. 0, u,; ?? , us; A, u,; tensionless (-)

1

I

1.0

r

1.5

1

2.0

I

J

2.5

of the ring subjected to a suddenly applied and conventional (- - -) Winkler foundation.

vertical

load

470

2. CELEP

Due to the character of the foundation, the initial values of the displacement are not equal for the two foundation models. In the above numerical results the oscillations appear as a result of a sudden change in the static load. The time variations of the displacements for the two foundation models when the load is applied suddenly are shown in Figure 8. In this case the ring, starting from rest, oscillates around its static equilibrium position. Since the load is uniformly distributed, the lower order vibration modes predominantly are excited and the oscillations are relatively simple in comparison with those shown in Figure 7. 4. CONCLUSIONS

Static and transient responses of an elastic ring supported on a elastic Winkler foundation have been studied and a number of numerical results generated in the presence and absence of lift-off, to investigate the effect of the tensionless character of the foundation. Singular and distributed external loads are considered. In the static case, due to the non-linearity of the lift-off, the results cannot be superposed when the foundation is assumed to be tensionless. Each loading case has to be treated separately. The tensionless nature of the foundation increases the contributions of the higher order vibration modes in the oscillations. A similar effect can be noticed when the load is singular. Due to the ring shape, the results for the tensionless and for the conventional Winkler foundations are significantly different, because a lift-off region in the ring always appears, even for very low foundation stiffness. It has also been shown that the extent of the lift-off region does not depend on the loading level, but varies with the relative rigidity of the foundation with respect to that of the ring and with the configuration of the loading. Moreover, it has been seen that for a given loading configuration the displacements vary linearly with the level of the load for the tensionless foundation model as well.

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flexural vibrations of circular rings. 5. C. P. FILIPICH,P. A. A. LAURA, M. ROSALESand 0. H. VALERGA DE GRECO 1987 Journal of Sound and Vibration 118, 166-169. Numerical experiments on in-plane vibrations of rings of non-uniform cross section. 6. S. S. FUO 1971 Aeronautical Journal 75, 417-419. Three-dimensional vibrations of a ring on an elastic foundation. 7. R. K. MITTAL 1976 International Journal of Engineering Science 14, 247-257. Flexure of a thin elastic ring due to a dynamic concentrated load. 8. Z. CELEP1988 Journalof Engineering Mechanics 114, 1723-1738. Circular plates on a tensionless Winkler foundation. 9. Z. CELEP 1988 Journal of Engineering Mechanics 114, 2083-2092. Rectangular plates resting on a tensionless elastic foundation. 10. Z. CELEP, D. TURHAN and R. Z. AL-ZAID 1988 Journal of Applied Mechanics 55, 624-628. Circular elastic plates on elastic unilateral edge supports. 11. Z. CELEP, D. TURHAN and R. Z. AL-ZAID 1988 International Journal of Mechanical Sciences 30, 733-741. Contact between a circularplate and a tensionlesssupport. 12. Z. CELEP 1988 Journal of Sound and Vibration 125, 305-312. On the time response of square

plates on unilateral

support.

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13. Z. CELEP, A. MALAIKA and M. ABU-HUSSEIN 1988 Journal of Sound and Vibration 128, 235-246. Forced vibrations of a beam on a tensionless foundation. 14. Z. CELEP and D. TURHAN 1990 Journal of Applied Mechanics (to appear). Axisymmetric dynamic response of circular plates on tensionless elastic foundation. 15. M. ABRAMOWITZ and I.A. STEGUN 1965 Handbook of Mathematical Functions. New York: Dover. 16. L. COLLATZ 1966 Numetical Treatment of Diflerential Equations. Berlin: Springer-Verlag.